With all new particles around TeV scale, various promising signatures can be probe by LHC [1]}. For simplicity, we consider a degenerate mass spectrum of the inert Higgs doublet \(\eta \) , i.e., \(M_{\eta ^\pm }=M_{\eta _R}=M_{\eta _I}\) .
One promising signature is originated from the pair production of \(\eta ^\pm \)  at LHC, i.e., \(pp\rightarrow \eta ^+\eta ^-\rightarrow \ell ^+N_1+\ell ^-N_1\) . Since \(N_1\)  is the DM candidate, this signature is just \(\ell ^+\ell ^-+{E}_T\)  The LHC limits are obtained with \(\ell =e,\mu \) , while there would be \(\tau \)  leptons in our scenario. Here, we directly take such limits as a conservative estimation, because for non-universal Yukawa structure, the \(\tau \)  final states can be suppressed under certain circumstance.. In the left panel of Fig. REF , theoretical production cross section at 13 TeV is shown, which is calculated by the help of MadGraph5_aMC@NLO [2]}. Meanwhile, exclusion region by LHC direct searches are shown in the right panel [3]}, [4]}, [5]}. Two viable regions are now allowed by direct search. One is the heavy scalar region \(M_\eta >700\)  GeV and the other is the degenerate region \(150~{\rm GeV}\lesssim M_{N_1}\lesssim M_{\eta }<700\)  GeV. Together with the parameter space in Fig. REF , we find that the heavy scalar region might be probe by both LHC and \(\mu -e\)  conversion when \(M_{N_1}\lesssim 200\)  GeV. For \(M_{N_1}\gtrsim 200\)  GeV or \(M_{\eta ^+}\gtrsim 800\)  GeV, only LHC has the ability to probe. Meanwhile future upgrade \(\mu \rightarrow 3e\)  and LHC searches with compressed mass spectra are both hopeful to probe the degenerate region. Note that the LFV exclusion limits depend on the choice of Yukawa coupling \(h^{\prime }\) , i.e., \(h^{\prime }=0.01\) , but the LHC searches do not.
